From the end of 19 to the beginning of the 20th century, mathematics has developed into a huge discipline, and the Department of Classical Mathematics has established a complete system: number theory, algebra, geometry and mathematical analysis. Mathematicians began to explore some basic questions, such as what is a number? What is a curve? What is integral? What is a function? ..... In addition, how to deal with these concepts and systems is also a problem.

There are two classic methods. One is the old axiomatic method, but with the development of non-Euclidean geometry and various geometries, many of its shortcomings are exposed. The other is the construction method or the generation method, which often has limitations, and many problems can be solved without construction. In particular, many problems involving infinity often rely on logic, reduction to absurdity and even intuition. However, what is reliable and what is unreliable cannot be determined without analysis.

The analysis and research of basic concepts have produced a series of new fields-abstract algebra, topology, functional analysis, measure theory and integral theory. The perfection of the method is to establish a new axiomatic method, which was first put forward by Hilbert in 1899 "Geometry Basis".

Axiomatization of elementary geometry

In the 1980s, after non-Euclidean geometry was generally recognized, the discussion on geometric basis began. At that time, it was clear that Euclid's system had many problems: first, the points, lines and surfaces in the original definition of Euclid geometry were not definitions; Secondly, Euclid's geometry uses many intuitive concepts, such as "between ……", and there is no strict definition. In addition, the independence, non-contradiction and completeness of the axiomatic system are not proved.

In 1980s, German mathematician Bass put forward an axiom system and some important concepts such as order axiom. But some axioms in his system are unnecessary and some necessary axioms are not, so his axiom system is not perfect. Moreover, he has no systematic axiomatic thought, and his purpose is to bring metric geometry into projective geometry in other aspects-by introducing ideal elements.

Since the late 1980s, piano and his students have also conducted a series of research. Piano's axiomatic system has limitations; His student Pierre's Geometry as a Deductive System (1899) has too few basic concepts (only "point" and "motion"), which makes the necessary definitions and axioms extremely complicated, so that the logical relationship of the whole system is extremely chaotic.

The publication of Hilbert's Fundamentals of Geometry marks the arrival of a new era of mathematical axiomatization. Hilbert's axiom system is the model of all subsequent axioms. Hilbert's axiomatic thought has a far-reaching influence on the later development of mathematical foundation. His book was reprinted many times and became a widely circulated classic document.

The difference between Hilbert's axiomatic system and Euclid and any axiomatic system after it is that he has no original definition, and the definition is embodied by axioms. He revealed the idea at 189 1. He said: "We can use tables, chairs and beer cups instead of dots, lines and noodles." Of course, he didn't mean to study tables, chairs and beer cases in geometry, but to abandon the intuitive meaning of points, lines and surfaces in geometry and only study the relationship between them, which is embodied by axioms. Geometry is a logical analysis of space without intuition.

Hilbert's axiom system includes twenty axioms, which he divides into five groups: the first group of eight axioms is related axioms (subordinate axioms); The second set of four axioms is the order axiom; The third group of five axioms; The fourth group is parallel axioms; The fifth group, two, is a continuous axiom.

After establishing the axiomatic system, Hilbert's first task is to prove the non-contradiction of the axiomatic system. This requirement is natural, otherwise, if contradictory results are deduced from this axiomatic system, then this axiomatic system is worthless. Hilbert proved the non-contradiction of his axiomatic system in the second chapter of Fundamentals of Geometry. This time, he can't put forward Euclidean model like non-Euclidean geometry, but he put forward arithmetic model.

In fact, from the perspective of analytic geometry, points can be interpreted as three arrays (which can be understood as coordinates (x, y, z)) and straight lines can be expressed as equations. It is not difficult to prove that this model satisfies all 20 axioms. Therefore, if there is any contradiction in the inference of axioms, it must be shown in the arithmetic of real number field. This changes the contradiction of geometric axioms into the contradiction of real number arithmetic.

Secondly, Hilbert considers the independence of axiomatic system, that is, axioms are not redundant. If an axiom cannot be deduced from other axioms, then it is independent of other axioms. If it is deleted from the axiomatic system, it will affect some conclusions. Hilbert's method of proving independence is to establish a model, so that all axioms except the axiom to be proved (such as parallel axiom) are established, and the negation of this axiom is also established.

Because these axioms are independent and not contradictory, we can add or delete axioms or negate them, thus obtaining new geometry. For example, if the parallel axiom is replaced by its negation, you will get non-Euclidean geometry; Archimedes axiom (to the effect that a short line segment can always exceed an arbitrarily long line segment after a finite number of repetitions) is replaced by non-Archimedes axiom, and non-Archimedes geometry is obtained. Hilbert discussed the properties of non-Archimedean geometry in detail in his book.

Hilbert's axiom of elementary geometry is not contradictory to real numbers, so it is natural to further consider the axiomatization of real number system and its non-contradiction, so the first problem is the axiomatization of arithmetic.

Axiomatization of arithmetic

Mathematics, as its name implies, is the science of studying numbers. Natural Numbers and Their Calculations-Arithmetic is the most obvious starting point of mathematics. Many people in history think that all classical mathematics can be derived from natural numbers. However, until the end of19th century, few people explained what number was. What is 0? 1 What is it? These concepts are considered as the most basic concepts, and whether they can be further analyzed is a concern of some mathematicians. Because once arithmetic has a foundation, other mathematics departments can also be firmly based on arithmetic.

What can be used as the basis of arithmetic? There are three ways in history: Cantor's cardinal number theory establishes natural numbers on the basis of set theory and extends them to infinity; Frege and Russell defined numbers completely through logical vocabulary, and established arithmetic on the basis of pure logic. Axiomatic method is used to define the number itself, the most famous of which is Piano's axiom.

Before the piano, there was Dai Dejin's axiomatic definition. His method is to be extended to rational numbers and real numbers, laying the foundation for mathematical analysis. They all notice that logic is the foundation, but they all have illogical axioms.

1888, Dai Dejin published "What is number, what is the purpose of number? , expounded his mathematical point of view. He regards arithmetic (algebra and analysis) as a part of logic, and the concept of number is completely independent of human representation or intuition of space and time. He said, "Numbers are the free creation of human mind. As a tool, they can make many things easier and more accurate. " . And the way to create is through logic. His definition is pure logical concepts-system, union and intersection of classes, mapping between classes, similarity mapping (different elements are mapped to different elements) and so on. Through axiomatic definition, Dai Dejin proved mathematical induction. But he can't directly define numbers from pure logical nouns.

1889, piano published his "arithmetic principle: a new discussion method", in which two things were obviously done: first, arithmetic was obviously based on several axioms; Second, axioms are expressed by new symbols. Later, piano, like Frege, described the sequence from 0, but his logarithmic concept, like Dai Dejin, considered ordinal numbers.

Piano's interest mainly lies in clearly expressing mathematical results, so does the symbol of mathematical logic he compiled (1894 published in On Mathematics), not for philosophical analysis. After Russell learned this set of symbols from the piano in 1900, it had a great influence on logic, philosophy and mathematics.

From 1894 to 1908, piano published the sequels of mathematical essays five times in a row, each time based on his five axioms (only 0 generation 1). But the piano has three other basic symbols besides logical symbols, namely, number, zero and succession. Therefore, unlike Frege and Russell, he didn't base the numbers entirely on logic.

His axiom system is also flawed, especially the fifth axiom involves all properties, so it is necessary to prove properties or sets. Someone changed it into a countable axiom sequence, so that the axiom system does not define a simple natural number. In 1934, Skram proved that there is a nonstandard model in the axiomatic system of piano, which destroyed the category of axiomatic system.

Axiomatization of other mathematical objects

In the wave of axiomatization from the end of 19 to the beginning of the 20th century, a series of mathematical objects were axiomatized, and these axiomatizations were generally carried out in mathematics. For example, the concepts of fields and groups introduced by solving algebraic equations were very specific at that time, such as permutation groups. It was not until the second half of19th century that the concept of abstract group gradually appeared and was described by axioms. There are four common reasons for a group, that is, the closed axiom, the addition (or multiplication) of two elements still corresponds to a unique element, the operation satisfies the associative law, and there are zero elements and inverse elements.

Groups are everywhere in mathematics, but the study of abstract groups did not begin until the end of the nineteenth century. Of course, it is closely related to mathematical logic. Rational number set, real number set and complex number set constitute the concrete model of abstract domain, and there are many axioms of domain. In addition, rings, posets, totally ordered set, lattices and Boolean algebras have all been axiomatized.

The other structure is topological structure. The topological space is also axiomatized from 19 14 to 1922. Hilbert space and Banach space in functional analysis were also axiomatized in the 1920s, which became the starting point of abstract mathematics research in the 20th century. In model theory, these mathematical structures become the models of logical sentence-making theory.